Earthquake　magnitude,earthquake　fault,and　source　dimensions

ABSTRACT

　Earthquakes　accompanied　by　faults　and　their　source　dimensions　were　investigated.The　magnitudes　of　these　earthquakes　are　related　to　the　length　of　faults　and　their　slip　displacements.The　dimensions　of　earthquake　source　volume　are　estimated　from　the　observational　dimensions　of　earthquake　fault.The　greater　the　earthquake　magnitude,the　larger　is　the　fault　length　at　the　earth’s　surface　and　its　upper　limit　exists　for　each　magnitude.Observed　upper　limit　is　somewhat　long　in　comparison　with　the　length　expected　from　the　crustal　strength.The　ratio　of　the　thickness　of　the　earth’s　crust　to　the　focal　depth　of　an　earthquake　is　related　to　the　fault　length.The　geometrical　situation　of　an　earthquake　source　dimension　with　respect　to　the　earth’s　crust　and　the　probability　of　fault　appearance　are　investigated.The　shallow　earthquakes　having　magnitude　greater　than　7.0　are　accompanied　by　a　surface　fault　when　the　crustal　thickness　is　forty　kilometers.

INTRODUCTION

　A　system　of　fractures　of　land　surface　has　been　generally　observed　in　the　epicentral　region　related　to　the　occurrence　of　a　great　earthquake　with　its　epicenter　on　land.Faulting　is　a　typical　event　occurring　simultaneously　with　an　earthquake　and　it　is　believed　to　have　sometimes　brought　about　the　release　of　a　large　amount　of　strain　energy　stored　in　the　earth’s　crust.Further,the　length　and　direction　of　a　fault　are　generally　known　to　be　related　to　the　magnitude　and　source　mechanism　of　the　pertaining　earthquake.It　is,therefore,important　to　see　the　nature　of　faulting.The　relationship　between　the　fault　length　and　the　earthquake　magnitude　was　investigated　in　a　previous　paper　[8].After　that,a　number　of　earthquakes　accompanied　by　fault　were　known　and　fault　data　were　extensively　corrected.In　the　present　paper,the　relationship　between　the　magnitude　of　earthquakes　and　dimensions　of　faults,and　source　dimensions　based　on　faulting　are　discussed　by　using　world　data.

DIMENSIONS　OF　EARTHQUAKE　FAULTS

　A　number　of　destructively　large　earthquakes　are　known　to　have　taken　place　in　Japan,but　these　have　not　always　accompanied　by　fault.For　instance,of　about　140　destructive　large　earthquakes　with　epicenters　on　land　since　1840,only　twelve　shocks　are　known　to　have　been　accompanied　by　surface　faults.The　earthquake　fault　data　collected　[14]　for　the　whole　world　are　given　in　Table1　which　includes　more　additional　and　corrected　data　than　those　in　a　previous　paper　[8]　From　Table1　the　number　of　faults　accompanying　land　earthquakes　totaled　sixty　four　during　the　period　from　1811　to　1964;twenty　faults　in　North　America,seven　in　New　Zealand,six　in　Turkey,three　in　India,four　in　Mongolia　and　some　in　other　countries.The　world　data　show　that　the　observed　relative　slip　displacement　varies　from　a　few　centimeters,in　a　very　small　earthquake,up　to　14.3meters,in　the　largest　observed　on　the　dip−slip　fault　of　the　Yakutat　earthquake　of　September　1899　in　Alaska.The　strike−slip　displacement　also　varies　with　the　size　of　the　earthquake,from　tens　of　centimeters　in　the　smallest　up　to　about　10meters　in　the　largest　observed　on　the　fault　of　the　outer　Mongolia　earthquake　of　April　1957.The　fault　where　the　slip　is　horizontal　is　designated　a　strike　slip　or　transcurrent　fault　and　the　fault　where　the　slip　is　vertical　or　perpendicular　to　the　line　formed　by　the　intersection　of　the　fault　with　the　horizontal　surface　is　known　as　a　dip−slip　fault.Combinations　of　dip−slip　and　strike−slip　faulting　also　occur,but　some　of　faults　are　predominant　in　the　former　and　some　are　predominant　in　the　latter.
　The　horizontal　extent　of　the　fault　varies　with　the　magnitude　of　the　earthquake,from　a　few　meters　in　the　smallest　up　to　an　order　of　1000kilometers　in　the　largest.The　longest　directive　observed　break　was　that　on　the　fault　of　Mongolia　earthquake　of　1905,in　which　the　trace　was　visible　on　the　ground　for　a　distance　of　about　700　kilometers.
　　A　large　number　of　the　great　earthquakes　of　the　world　occur　along　the　margins　of　the　circum−Pacific　continents.Some　of　these　earthquakes　are　accompanied　by　faults　which　intersect　the　surface　offshore,and　consequently　the　extents　of　the　faults　cannot　be　observed　directly.Fault　extents　can　be　estimated　indirectly　from　the　lengths　of　the　distribution　patterns　of　after−shocks　parallel　to　the　fault　and　by　a　method　based　upon　the　spectra　of　surface　waves　such　as　Rayleigh　and　Love　waves　[4,5,13].In　the　great　Chilean　earthquake　of　20　May　1960,a　length　of　fault　break　of　about　1000kilometers　was　estimated　by　both　methods　[13].

ESTIMATION　OF　SPEED　OF　SURFACE　FAULT　FORMATION

　The　speed　of　surface　fault　formation　will　be　estimated　as　follows　after　H.Benioff　[3]:The　horizontal　speed　of　rupture　propagation　is　found　to　be　about　3　to　4kilometers　per　second　for　strike−slip　faulting　and　somewhat　less　for　dip　slip　fault　formation　[4,5].Strike−slip　faulting　proceeds　faster　because　it　is　propagated　from　point　to　point　by　compressional　stresses,while　dip−slip　faulting　is　maintained　by　the　smaller　shearing　stresses.
　The　duration　of　slip　at　any　one　point　is　not　accurately　known　but　is　estimated　to　vary　from　a　fraction　of　a　second,in　the　smallest　earthquakes,to　about　10　seconds　in　the　largest.The　duration　of　the　earthquake　at　the　source　is　totally　equal　to　the　length　of　rupture　divided　by　the　speed　of　rupture　propagation.In　the　largest　earthquakes,with　1000kilometers　of　surface　fault,the　duration　of　the　source　is　thus　about　280　seconds,approximately.In　the　case　of　the　Kern　County　earthquake　of　1952　the　time　required　for　the　faulting　to　reach　the　surface　from　the　focus　was　also　estimated　by　Benioff　[2,3]　and　was　greater　than　about　3　seconds.The　time　[2]　required　for　the　faulting　to　progress　horizontally　from　the　focus　to　the　end　point　in　the　vicinity　of　Caliente,California　was　greater　than　16　seconds.
　The　accumulation　rate　of　elastic　strain　is　approximately　estimated　by　Whitten　and　Claire　[19]　for　the　San　Andreas　fault　by　data　from　triangulation　surveys　of　the　strained　zone.They　found　that　the　two　opposed　fault　blocks　are　moving　relative　to　each　other　at　a　rate　of　about　5　centimeters　per　year.If　this　rate　is　constant　and　if　each　break　of　the　fault　occurs　at　the　same　value　of　strain,earthquakes　on　any　given　segment　should　repeat　at　intervals　of　about　125years　[3].However,there　are　many　unknown　factors　to　estimate　the　repetition　of　earthquakes　with　fair　accuracy.Especially　the　actual　vertical　extent　of　the　slip　is　not　known　even　approximately　for　any　earthquake,though　it　is　generally　assumed　that　the　break　in　shallow　earthquakes　does　not　extend　below　the　Moho　discontinuity,but　is　restricted　to　near　the　surface　[7].

MAGNITUDES　OF　EARTHQUAKES　AND　DIMENSIONS　OF　FAULTS

　The　magnitude　of　earthquakes　which　have　been　accompanied　by　fault　is　generally　larger　than　5.6　which　is　also　the　smallest　magnitude　of　earthquake　accompanied　by　crustal　deformation　or　tsunami,as　shown　in　Figs.1　and　2.Earthquakes　with　a　magnitude　greater　than　7.3　are　always　accompanied　by　fault　in　Japan,and　shallow　submarine　earthquakes　with　a　magnitude　greater　than　7.3　are　also　always　accompanied　by　tsunami,as　already　pointed　out　[9].
　Since　the　maximum　strain　of　the　earth’s　crust　is　of　the　order　of　10^−4　[18],the　elastic　strain　energy　stored　in　the　crust　just　before　an　earthquake　may　be　derived　from　the　earthquake　fault　data　such　as　maximum　length　at　the　earth’s　surface　and　maximum　displacement　at　the　fault.The　relationship　between　the　length　l　of　surface　fault　and　the　earthquake　magnitude　M　is　shown　in　Fig.3.This　relationship　may　be　determined　by　the　least　squares　method　such　as
　　　M＝（0.76±0.08）log　l＋（6.07　±0.13）,（1）
where　l　is　measured　in　kilometers.This　relationship　shows　generally　that　the　greater　the　earthquake,the　larger　is　the　fault　length　at　the　surface.It　is　found　that　the　exponent　of　l　is　about　0.8.Considering　the　scattering　of　the　data　in　Fig.3,we　may　take　the　exponent　of　l　as　unity　for　the　range　of　earthquake　magnitude　from　5.6　to　8.5.（1）is　the　revised　relationship　obtained　from　more　additional　data　of　land　earthquakes　than　those　in　a　previous　paper　[8].
　The　relationship　between　the　fault　length　1　and　the　maximum　displacement　D　is　shown　in　Fig.4,which　may　be　determined　by　the　least　squared　method　such　as
　　　logl＝（0.93±0.22）logD＋（1.25±0.12）,（2）
　　　logD＝（0.55±0.07）M−（3.71±0.50）,（3）
where　D　is　the　largest　value　of　either　the　maximum　dip−slip　D_v　or　maximum　strike−slip　D_h,in　meters.Fig.5　shows　the　relationship　between　the　earthquake　magnitude　and　the　largest　displacement　D.Fig.6　shows　the　relationship　between　the　maximum　dip−slip　D_v,and　the　maximum　strike　slip　D_h.Combining　
（1）or（3）with　Gutenberg−Richter’s　formula　of　seismic　wave　energy　E_s:
　　　logE_s＝11.8＋1.5　M,（4）
we　obtain　the　following　relation　respectively:
　　　E_s＝7.8　×　10^20　×　l^1.2,（5）
　　　E_s＝6.6×10^21×D^2.7.　（6）
Here,the　significance　of　the　exponent　of　l　in（5）or（6）is　considered　to　be　related　to　the　spatial　shape　of　rock　which　releases　the　elastic　strain　energy.If　only　a　certain　range　of　length　of　ground　breakage　contributed　to　the　release　of　strain　energy,the　exponent　of　l　should　be　close　to　unity.Seeing　that　the　exponent　of　l　in（5）is　nearly　equal　to　unity,the　fault　length　is　considered　to　contribute　to　the　wave　energy　of　the　pertaining　earthquake.As　seen　in（5）,the　wave　energy　of　an　earthquake　released　would　be　approximately　proportional　to　the　horizontal　extent　of　surface　break　for　large　earthquakes　which　had　foci　in　the　crust　as　reported　in　a　previous　paper　[8].（6）shows　that　the　released　earthquake　energy　would　be　approximately　proportional　to　the　cube　of　the　slip　displacement.
　The　extent　perpendicular　to　the　earthquake　fault　of　the　zone　of　strained　rock　is　considered　to　be　proportional　to　the　displacements　of　the　ground　at　the　fault　as　a　first　approximation.Assuming　that　the　elasticity　of　rocks　near　the　earth’s　surface　is　the　same　in　both　the　horizontal　and　vertical　directions,the　area　at　the　earth’s　surface　of　the　zone　of　strained　rock　is　proportional　to　the　product　√（D^2_v,＋　√D^2_hl）in　which　D_v　and　D_h　represent　dip−slip　and　strike−slip,respectively.The　relationship　between　M　and　√D^2_v＋　D^2_hl,（＝Rl）or　M　and　D_vl　or　D_hl　for　each　shock　in　Table1　can　also　be　determined　by　the　method　of　least　squares　to　the　linear　relation,as　shown　in　Figs.7　and　8.
　We　get　the　relations　as
　　M＝（6.16±0.30）＋（0.58±0.15）log　Rl,（7）
　　M＝（6　44±0.34）＋（0.53±0.20）log　Dl,（8）
in　which　l　and　R　are　measured　in　meters.
　Combining（4）and（7）,or（4）and（8）to　eliminate　M　as　before,we　get
　　log　E_s＝21.0＋0.9　log　Rl,
　　log　E_s＝21.4＋0.8　log　Dl　（9）
or　　　E_s＝1.0　×　10^21　×（Rl）^0.9,
　　　　E_s＝2.8　×　10^21　×（Dl）^0.8.（10）
Once　the　displacements　and　the　length　of　earthquake　faults　are　determined,the　maximum　released　energy　of　earthquakes　can　be　estimated　from（8）as　before.However,from　these　equations　the　exponent　of　R1　or　D1　is　close　to　unity.It　may　be,therefore,concluded　that　the　energy　released　in　the　larger　shallow　earthquakes　is　approximately　proportional　to　the　area　at　the　earth’s　surface　of　the　zone　of　strained　rock,as　already　pointed　out　[1,8,16].

AFTERSHOCK　VOLUME,EARTHQUAKE　MAGNITUDE　AND　FRACTION　OF　ENERGY　RELEASED　AS　SEISMIC　WAVES

　In　connection　with　the　energy　of　earthquakes,aftershock　volume　was　investigated　by　using　the　aftershock　area　and　the　epicentral　depth　of　aftershocks　including　main　shock,as　shown　in　Fig.9.The　data　[10]　in　this　figure　may　be　expressed　by　the　least　squares　method　as
　　　logV＝（1.06±0.10）M−（2.78±0.74）,（11）
where　V　is　the　volume　of　aftershock　activity　in　km^3　and　M　is　from　6　to　8.5.If　the　aftershock　volume　is　assumed　to　be　an　earthquake　volume　V　where　the　strain　energy　is　stored,the　energy　of　the　released　seismic　waves,E_s　is
　　　E_s＝1/2efx^2−V,（12）
in　which　ex^2　V/2　is　the　potential　energy　of　a　volume　V　of　the　crustal　material　having　an　average　elastic　constant　e　strained　an　ultimate　amount　x　immediately　before　an　earthquake,and　f　the　fraction　of　the　energy　E_s　released　as　seismic　waves.
　Assuming　reasonable　values　of　e＝5×10^11　C.G.S.and　x＝10^（−4）for　an　understanding　of　the　nature　of　the　problem　and　combining（4）,（11）and（12）the　following　relationship　may　be　approximately　derived　as　[10]
　　　0.5M　＝　logf＋4.5.（13）
Thus,the　fraction　of　the　energy　released　as　seismic　waves　may　depend　upon　the　magnitude　M　of　an　earthquake.We　may　calculate　the　traction　of　conversion　of　elastic　strain　energy　to　wave　energy　from（13）.For　instance,
　　f■1　for　the　earthquake　magnitude　M＝9,
　　f■1/3　for　M＝8,
　　f■1/10　for　M＝7,
　　f■1/30　for　M＝6.
　It　is,therefore,concluded　that　the　greater　the　magnitude　of　an　earthquake,the　more　often　will　the　efficiency　of　conversion　of　earthquake　strain　energy　to　seismic　wave　energy　will　approach　unity.
　In　connection　with　the　fraction　of　conversion　of　strain　energy　to　wave　energy,Bullen　[6]　obtained　the　earthquake　energy　E_s　based　on　elastic　theory　as
　　　E_s＝S^2V/12qμ’　（14）
where　V　is　the　volume　of　the　heavily　strained　zone　in　an　earthquake,S　the　Mises　strength,μ　the　rigidity,q　the　ratio　of　distorsional　strain　energy　to　seismic　energy.In　this　case　q　is　corresponding　to　1/f　in（12）.He　gave　relevant　values　of　S＝10^9　dyne/cm^2,μ＝0.4×10^12　dyne/cm^2,q＝2　for　the　greatest　earthquake.
From　q＝2,we　get　f＝1/2　which　is　equal　to　f　for　the　case　of　M＝8.4.

SOURCE　DIMENSIONS　AND　PROBABILITY　OF　FAULT　APPEARANCE

　The　probability　of　fault　appearance　and　source　dimensions　are　investigated.For　simplicity,we　take　the　same　consideration　as　that　of　Otsuka　[11]　and　assume　that　the　earthquake　source　is　spherical　in　shape　with　radius　a　and　that　the　occurrence　of　fracture　is　restricted　within　the　earth’s　crust　of　thickness　H.Then　if　the　following　conditions　are　satisfied,the　fault　will　appear　at　the　earth’s　surface:
　　　d＜H−a,（15）
　　　d＜a,（16）
where　d　is　the　depth　of　the　center　of　the　source.If　the　occurrences　of　earthquakes　satisfy　conditions（15）and（16）,the　probability　P（M）of　an　earthquake,having　the　spherical　source　of　radius　a　and　the　magnitude　M,accompanied　by　surface　fault　is　expressed　by
　　　【数式】　（17）
where　n（M）is　the　total　number　of　earthquakes,n’（M）the　number　of　earthquakes　accompanied　by　faults.The　elastic　strain　energy　contained　in　a　spherical　source　volume　is　expressed　as
　　　E＝2/3πex^2a^3.（18）
If　the　values　of　e　and　x　are　taken　as　mentioned　in　Section　5,then　E　in　ergs
becomes
　　　E＝10^4×a^3（a　in　cm）.（19）
Combining（19）,（4）,（12）and（13）,the　following　relation　can　be　obtained　as
　　　loga（M）＝0.33M−1.0（a　in　km）.（20）
From（20）the　probability　that　an　earthquake　of　magnitude　M　is　accompanied　by　surface　fault　can　be　obtained.If　H　is　assumed　to　be　50km,the　calculation　shows　that　the　percentage　of　number　n’（M）of　earthquakes　with　fault　appearance　against　the　total　number　n（M）of　earthquakes　of　that　magnitude　is　100　per　cent　for　M＝7.3.The　calculated　probability　for　the　case　of　H＝40km　is　100　per　cent　for　M＝7.0,as　given　in　Table2.These　calculations　are　close　to　the　observed　data　[8,16]　that　shallow　earthquakes　having　magnitude　greater　than　6.5　or　7.3　are　almost　accompanied　by　surface　faults.Further,the　size　of　the　spherical　source　calculated　from（20）is　also　given　in　Table2.
From　the　conditions（15）and（16）,we　can　derive　the　following　relation:
　　　1＜2a.　（21）
Then,if　the　value　of　a　is　given,the　longest　fault　length　l_m,for　certain　earthquake　magnitude　is　determined　by
　　　l_m＝2a.（22）
The　number　of　earthquakes　of　which　the　observed　fault　length　l　is　larger　than　l_m＝2a　is　given　in　Table3　where　the　number　n　of　earthquakes,of　which　the　observed　fault　length　l　is　smaller　than　l_m,is　also　given.
　These　results　are　considered　to　depend　on　the　certain　limit　of　fault　length　at　the　earth’s　surface.The　greater　the　magnitude　of　the　earthquake,the　larger　becomes　the　difference　between　the　value　observed　and　that　calculated　by（22）.This　suggests　that　the　source　volume　can　no　longer　be　spherical　when　the　earthquake　magnitude　exceeds　a　certain　value　which　is　different　from　the　thickness　of　the　earth’s　crust.

LIMIT　OF　FAULT　LENGTH　AND　SOURCE　DIMENSIONS

　The　fault　length　of　each　earthquake　depends　upon　the　degree　of　shocks,the　size　of　earthquake　source,focal　depth,source　mechanism,and　crustal　strength.If　the　conditions　of　the　earthquake　occurrences　would　be　the　same,an　upper　limit　of　fault　length　would　exist　for　each　earthquake　magnitude　because　of　the　average　energy　density　being　almost　the　same.
　The　relationship　between　the　maximum　fault　length　l_max　or　minimum　fault　length　l_mja　and　the　earthquake　magnitude　M　may　be　obtained　by　the　chain　and　dotted　lines　drawn　in　Fig.10　where　the　ratio　of　the　thickness　of　the　earth’s　crust　to　the　focal　depth　of　an　earthquake　is　given.As　seen　in　Fig.10,the　large　values　of　this　ratio　are　distributed　near　the　upper　limit　line　and　the　maximum　fault　length　l_max　for　earthquakes　of　magnitude　M　increases　with　the　increase　in　M.The　lines　for　l_max　and　l_min　are　approximately　expressed　by
　　　M＝2logl_max＋3.5,（23）
and　　M＝2logl_min＋7.0,（24）
respectively.l　in（23）or（24）is　measured　in　kilometers.
　Eliminating　M　from（23）,（24）and（4）,the　seismic　wave　energy　E_s　can　be　obtained　as
　　　logE_s＝2.1＋3　log　l_max,（25）
or　　　E_s＝10^2　×　l^3_max,
and　　　logE_s＝7.3　＋　3logl_min,（26）
or　　　E_s＝2×10^7×l^3_min
in　which　l　is　measured　in　centimeters.
　Combining（22）,（23）,（4）and（12）,we　get　the　approximate　value　of　the　fraction　of　energy　conversion,as　f＝1/10　which　corresponds　to　the　average　value　of　f　derived　from（13）for　6＜M＜　8.At　any　rate,we　get　the　maximum　value　of　the　source　dimension　of　an　earthquake　from（23）and（22）as　follows:
【表】
As　already　mentioned　in　the　former　section,if　the　earthquake　magnitude　exceeds　a　certain　value,source　volume　may　be　not　spherical,because　the　dimension　of　the　sphere　is　larger　than　the　crustal　thickness.Thus,the　volume　must　be　flattened　between　the　earth’s　surface　and　the　bottom　of　the　earth’s　crust.This　will　explain　why　extraordinary　long　faults　sometimes　are　observed　in　the　large　earthquakes　and　also　why　the　minimum　length　of　the　fault　exists　for　earthquakes　greater　than　M＝7.3.

CONCLUDING　REMARKS

　The　relationship　between　the　observational　fault　dimensions　and　the　magnitude　of　earthquakes　is　investigated.The　observed　relative　slip　displacement　varies　from　a　few　centimeters　up　to　about　14meters.Some　of　faults　are　predominant　in　strike　slip　displacement　and　some　are　predominant　in　dip−slip　displacement.The　horizontal　extent　of　the　fault　varies　with　the　magnitude　of　an　earthquake　from　a　few　meters　up　to　an　order　of　1000　kilometers.The　minimum　magnitude　of　the　earthquake　accompanied　by　visible　fault　was　5.6.Earthquakes　with　a　magnitude　greater　than　7　are　almost　accompanied　by　a　fault.This　may　well　be　explained　by　the　calculation　of　the　probability　of　fault　appearances　based　on　the　assumption　that　the　mantle　of　the　earth　does　not　concern　with　fracturing.Although　this　may　not　actually　be　the　case,this　assumption　seems　a　reasonable　and　simple　working　hypothesis.
　The　source　dimension　of　an　earthquake　is　estimated　by　using　the　fault　length.It’s　upper　limit　is　identified.with　the　maximum　linear　dimension　of　the　earthquake　source　volume.The　greater　the　magnitude　of　an　earthquake,the　larger　becomes　the　difference　between　the　observed　value　and　the　calculated　maximum　length.Observed　length　is　comparatively　long　in　comparison　with　the　length　expected　from　the　mechanical　strength　of　the　earth’s　crust.Further　investigation　will　be　necessary　for　fracturing　of　the　earth’s　crust　and　the　mantle.

REFERENCES

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